We have conducted an H I 21 cm emission-line survey using the Parkes 20 cm multibeam instrument and the Australia Telescope Compact Array (ATCA) of six loose groups of galaxies chosen to be analogs to the Local Group. The goal of this survey is to make a census of the H I-rich galaxies and high-velocity clouds (HVCs) within these groups and compare these populations with those in the Local Group. The Parkes observations covered the entire volume of each group with a rms MH I sensitivity of (4-10) × 105 M⊙ per 3.3 km s-1 channel. All potential sources detected in the Parkes data were confirmed with ATCA observations at ∼2' resolution and the same MHI, sensitivity. All the confirmed sources have associated stellar counterparts; no starless H I clouds - HVC analogs-were found in the six groups. In this paper, we present a description of the survey parameters, sensitivity, and completeness. Using the population of compact HVCs (CHVCs) around the Milky Way as a template, coupled with the detailed knowledge of our survey parameters, we infer that our nondetection of CHVC analogs implies that, if similar populations exist in the six groups studied, the CHVCs must be clustered within 90 kpc of group galaxies, with average MH I, ≲ 4 × 105 M⊙ at the 95% confidence level. The corollary is that the same must apply to Milky Way CHVCs. This is consistent with our previous results from a smaller sample of groups, and in accordance with recent observational and theoretical constraints from other authors. These results confirm that there is very little neutral matter around galaxies, and that any substantial reservoir of baryons must be in other phases.